Base field 3.3.1620.1
Generator \(a\), with minimal polynomial \( x^{3} - 12 x - 14 \); class number \(1\).
Weierstrass equation
This is a global minimal model.
Mordell-Weil group structure
\(\Z \oplus \Z\)
Mordell-Weil generators
| $P$ | $\hat{h}(P)$ | Order |
|---|---|---|
| $\left(-67 a^{2} + 93 a + 675 : -1705 a^{2} + 2371 a + 17161 : 1\right)$ | $0.10197829462292110456679008894485744001$ | $\infty$ |
| $\left(-\frac{33883}{49} a^{2} + \frac{6731}{7} a + \frac{341083}{49} : \frac{15298105}{343} a^{2} - \frac{3039725}{49} a - \frac{153982681}{343} : 1\right)$ | $1.8674495367019167155765352141116092184$ | $\infty$ |
Invariants
| Conductor: | $\frak{N}$ | = | \((a^2-a-8)\) | = | \((a+2)\cdot(a+1)^{2}\) |
|
| |||||
| Conductor norm: | $N(\frak{N})$ | = | \( 18 \) | = | \(2\cdot3^{2}\) |
|
| |||||
| Discriminant: | $\Delta$ | = | $-576$ | ||
| Discriminant ideal: | $\frak{D}_{\mathrm{min}} = (\Delta)$ | = | \((-576)\) | = | \((a+2)^{18}\cdot(a+1)^{6}\) |
|
| |||||
| Discriminant norm: | $N(\frak{D}_{\mathrm{min}}) = N(\Delta)$ | = | \( -191102976 \) | = | \(-2^{18}\cdot3^{6}\) |
|
| |||||
| j-invariant: | $j$ | = | \( \frac{109503}{64} \) | ||
|
| |||||
| Endomorphism ring: | $\mathrm{End}(E)$ | = | \(\Z\) | ||
| Geometric endomorphism ring: | $\mathrm{End}(E_{\overline{\Q}})$ | = | \(\Z\) (no potential complex multiplication) | ||
|
| |||||
| Sato-Tate group: | $\mathrm{ST}(E)$ | = | $\mathrm{SU}(2)$ | ||
BSD invariants
| Analytic rank: | $r_{\mathrm{an}}$ | = | \( 2 \) |
|
|
|||
| Mordell-Weil rank: | $r$ | = | \(2\) |
| Regulator: | $\mathrm{Reg}(E/K)$ | ≈ | \( 0.096843165879431886091696811443252857005 \) |
| Néron-Tate Regulator: | $\mathrm{Reg}_{\mathrm{NT}}(E/K)$ | ≈ | \( 0.8715884929148869748252713029892757130450 \) |
| Global period: | $\Omega(E/K)$ | ≈ | \( 28.306529904916538050107400395234687372 \) |
| Tamagawa product: | $\prod_{\frak{p}}c_{\frak{p}}$ | = | \( 4 \) = \(2\cdot2\) |
| Torsion order: | $\#E(K)_{\mathrm{tor}}$ | = | \(1\) |
| Special value: | $L^{(r)}(E/K,1)/r!$ | ≈ | \( 2.4518878662338783115267664151299953210 \) |
| Analytic order of Ш: | Ш${}_{\mathrm{an}}$ | = | \( 1 \) (rounded) |
BSD formula
$$\begin{aligned}2.451887866 \approx L^{(2)}(E/K,1)/2! & \overset{?}{=} \frac{ \# ะจ(E/K) \cdot \Omega(E/K) \cdot \mathrm{Reg}_{\mathrm{NT}}(E/K) \cdot \prod_{\mathfrak{p}} c_{\mathfrak{p}} } { \#E(K)_{\mathrm{tor}}^2 \cdot \left|d_K\right|^{1/2} } \\ & \approx \frac{ 1 \cdot 28.306530 \cdot 0.871588 \cdot 4 } { {1^2 \cdot 40.249224} } \\ & \approx 2.451887866 \end{aligned}$$
Local data at primes of bad reduction
This elliptic curve is not semistable. There are 2 primes $\frak{p}$ of bad reduction.
| $\mathfrak{p}$ | $N(\mathfrak{p})$ | Tamagawa number | Kodaira symbol | Reduction type | Root number | \(\mathrm{ord}_{\mathfrak{p}}(\mathfrak{N}\)) | \(\mathrm{ord}_{\mathfrak{p}}(\mathfrak{D}_{\mathrm{min}}\)) | \(\mathrm{ord}_{\mathfrak{p}}(\mathrm{den}(j))\) |
|---|---|---|---|---|---|---|---|---|
| \((a+2)\) | \(2\) | \(2\) | \(I_{18}\) | Non-split multiplicative | \(1\) | \(1\) | \(18\) | \(18\) |
| \((a+1)\) | \(3\) | \(2\) | \(I_0^{*}\) | Additive | \(-1\) | \(2\) | \(6\) | \(0\) |
Galois Representations
The mod \( p \) Galois Representation has maximal image for all primes \( p < 1000 \) except those listed.
| prime | Image of Galois Representation |
|---|---|
| \(3\) | 3B.1.2 |
Isogenies and isogeny class
This curve has non-trivial cyclic isogenies of degree \(d\) for \(d=\)
3.
Its isogeny class
18.1-c
consists of curves linked by isogenies of
degree 3.
Base change
This elliptic curve is a \(\Q\)-curve. It is the base change of the following elliptic curve:
| Base field | Curve |
|---|---|
| \(\Q\) | 162.a2 |